Spare Parts are used to complete the Old Car. When you have completed the basic repairs to the car itself, (clicking on the car, using 100 coins per click), this is one of the items required to complete Mission V for the Old Car. This mission includes Spare Parts, Jumper Cables, Car Wax, and Spark Plugs.
Spare Parts are usually dropped by Plumber Zombies.
By clicking on the car while this mission is active, it will show you how many you need of each item. It will also permit you to ask your neighbors for the part. If you ask your neighbors, it has been suggested that you let them know (in the personal message option) that the parts do NOT come from their personal inventory.